Assignments

Supplementary problems are recommended but not required. Answers to supplementary problems will be given out with the answers to the assigned problems. You may and should work with others on these problems, but write out the final draft of the solutions by yourself. If you get stuck on a problem, talk to me before it is time to turn the solutions in.

Assignment 1:

Due Feb. 15, discussion Feb. 17

A. Text problems 1.1, 1.3ab, 1.6 [the best answer I ever got for this one was six words and a punctuation mark].

B. [Modified version of 1.11, 1.12, and 1.13. Challenging, but if you can work this problem you definitely understand probability distributions.]
(a) The needle on a broken car speedometer is free to swing, and bounces perfectly off the pins at either end, so that if you give it a flick it is equally likely to come to rest at any angle between 0 and p. What is the probability density r(q)? [r(q) dq is the probability that the needle will come to rest between q and q + dq.]

(b) Where does r (q) vanish?

(d) Calculate <q > for this distribution.

(e) Suppose that we are interested in the x-coordinate of the needle point -- that is, the "shadow" or "projection" of the needle on the horizontal line. What is the probability density r(x)? [Hint: You know the probability r(q); the question is what interval dx corresponds to the interval dq ?]

(f) Calculate <x> and <x²> for this distribution.

(g) A needle of length l is dropped at random onto a sheet of paper ruled with parallel lines a distance l apart. What is the probability that the needle will cross a line? [Yes, this part is related to the others.]

(h) Write out a quick summary, with no or few mathematical expressions, of the logic of your solution to part (g).

Assignment 2:

Due Feb. 22, discussion Feb. 24

A. Text problems 1.5, 1.7, 1.8, 1.9, 1.14, 1.15.
(Supplementary problem: Text problem 1.4)
Note on Gaussian integrals: ò _-¥^¥ exp [-x²/(2a)] dx can be looked up in a table of integrals. Then ò _-¥^¥ x ²ⁿ exp [-x²/(2a)] dx can be found by differentiating n times with respect to 1/a. Why don't you need to worry about the odd powers of x times the Gaussian?

B. Explain briefly without explicit mathematical expressions the logic of your solution to 1.14a.

Assignment 3:

Due Mar. 1, discussion Mar. 3
Text problems 2.1, 2.2, 2.5, 2.6, 2.36
(Supplementary problem: 2.4)

Assignment 4:

Due Mar. 8, discussion Mar. 10
Text problems 2.11ac [n=0 only], 2.12 [x, x², and p² only; do not check Uncertainty Principle], 2.16, 2.21 , 2.24, 2.42.

Assignment 5:

Discussion Mar. 17
Text problems 2.29 [plus: explain the logic of your solution], 2.35 , 2.46 , 2.47

Assignment 6:

Due Mar 24, discussion Mar 24
This assignment is simply a review of matrix algebra. You should know pretty quickly whether you remember what you should do. My notation is the best I can do and still post the page in html.

1. Suppose A and B are the matrices

        (  0   3 )                ( 1  0 )
    A = (        )    and    B =  (      )
        (-2i   2 )                ( 3  2 )  .

and ^h.c. means Hermitian conjugate. Compute [A, B] , A^*, A^h.c., Tr (B), det (B), and B^-1. Does A have an inverse?

2. Now include the column matrices a and b

         ( 2i )            ( 1-i )
     a = (    )   and  b = (     )
         ( 2  )            (  0  )  .

and let ^tr indicate a transposed matrix. Find A a, a^h.c.b, a^tr B b, and a b^h.c..

3. Prove that the product of two unitary matrices is unitary. Under what conditions is the product of two Hermitian matrices Hermitian? Is the sum of two unitary matrices unitary? Is the sum of two Hermitian matrices Hermitian? Don't forget that (A B)^h.c. = B^h.c. A^h.c. with the factors on the right in the reverse order of those on the left.

Assignment 7:

Due Apr. 7, discussion Apr. 7
A. Show that exp(-x²/2) is an eigenfunction of the operator Q = (d²/dx²) - x², and find its eigenvalue.

B. Text problems A.12 (p. 446) , 3.11, 3.38.

Assignment 8:

Due Apr. 12, discussion Apr. 14
4.2, 4.3 (also show that your solutions satisfy the differential equation), 4.5 (Y_ll only), 4.7, 4.9 [bonus credit for 4.9 given through Friday].

Assignment 9:

Due Apr 19, discussion Apr. 21
Text problems 4.18, 4.20, 4.22, 4.24 , 4.13, 4.14 , 4.16

Assignment 10:

Due Apr. 25, discussion Apr. 27
A. Text problems 4.27ab, 4.29, 4.31, 4.32a, 4.38.
B. Calculate the commutator [p_i, L_j], where p_i is the ith component of the momentum and L_j is the jth component of the angular momentum. Can a plane wave have definite angular momentum? You might want to use the relation S_i e_ijk e_imn = d_jmd_kn - d_jnd_km

Assignment 11:

Due May 3, discussion May 5
Text problems 5.2abc , 5.4 , 5.5 , 5.7 , 5.33

Assignment 12:

Due (and discussion) May 17
Text problems 6.2, 6.3, 6.7

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Last revised 2006/02/06